I am having a bit of a trouble understanding linear combinations.
Say, for example, I have a vector
$$Z=
\begin{bmatrix}
4 \\
-5 \\
-4 \\
\end{bmatrix}
$$
Would this vector be a linear combination of
$$X=
\begin{bmatrix}
1 \\
1 \\
-1 \\
\end{bmatrix}
$$
and $$Y=
\begin{bmatrix}
2 \\
-1 \\
-2 \\
\end{bmatrix}
$$
Best Answer
Just try to write $Z$ as $aX+bY$. In other words, solve the system$$\left\{\begin{array}{l}a+2b=4\\a-b=-5\\-a-2b=-4.\end{array}\right.$$Does it have solutions? Then $Z$ is a linear combination of $X$ and $Y$. Otherwise, it is not.